Huaihui Chen, Nanjing Normal University.

*
*

Alexander Strohmaier, Loughborough University, UK.

* The semi-classical limit of smooth and continuous quantum systems
is well covered in the literature and leads to classical mechanics,
thus establishing Bohr's correspondence principle. The situation changes when
one deals with Quantum mechanical systems with discontinuous metrics. Ray splitting
occurs and the semi-classical limit is not described by a classical flow any more.
I will explain to what extent the semi-group emerging from a ray-splitting flow
replaces the classical dynamics in this context. I will show that one can prove Quantum ergodicity
in this context under very natural assumptions.
(joint work with D. Jakobson and Y. Safarov)
*

Susanna Dann, University of Missouri at Columbia.

* The classical Busemann-Petty problem asks whether origin-symmetric convex bodies in R^n with corresponding smaller
central hyperplane sections necessarily have smaller volume. The answer is affirmative for n <= 4 and negative for n >= 5.
We study this problem in the complex hyperbolic n-space and prove that the answer is affirmative for n <= 2 and negative for n >= 3.
*

Martin Klimes, DMS, Université de Montreal.

* We consider the question of existence and domain of bounded solutions of systems of analytic ODEs near a singular point.
In case of a multiple singularity, these solutions often possess a divergent asymptotic expansion. A standard method for obtaining such a solution is then to take the Borel-Laplace sum of the divergent series.
We explain on an example how to generalize the Borel-Laplace method to study these solutions under small analytic perturbations when a double singularity splits into two simple singularities.
The example we consider is that of analytic parameter dependent systems of ODEs
$(x^2-\epsilon)\frac{dy}{dx}=My+f(x,y,\epsilon)$,
under a generic condition that the matrix $M$ of the linear part is invertible.
Our approach allows for a unified treatment in the spirit of resurgent analysis for all values of the perturbation parameter $\epsilon$ in a neighborhood of 0.
*

Maamoun Turkawi, Montreal.

* Soit \Gamma un graphe vérifiant des hypothèses géométriques convenables. On va donner diverses caractérisations équivalents pour les espaces de Sobolev et Hardy-Sobolev sur
\Gamma, en termes de fonctions maximales, fonctions de type Hajlasz ou décompositions atomiques.
*

Javad Mashreghi, Université Laval.

* Let $\varphi: $\mathbb{D} \to $\mathbb{D}$ be analytic and consider the (composition) mapping $C_\varphi(f) = f \circ \varphi$ on
the family of holomorphic functions. Then the Hardy space $H^2$ is closed under all $C_\varphis$’s.
This is the classical {\em subordination principle of Littlewood} stated in the language of operator theory.
One can ask a similar question for any function space on $\mathbb{D}$. We discuss the composition operators on model spaces $K_\Theta$ and
de Branges-Rovnyak spaces $\mathcal{H}(b)$. In both cases, in particular
the latter, the question is still wide open and just partial results are available.
*

Thierry Daude, Université de Cergy-Pontoise.

*
In this talk, we first describe a class of axisymmetric, electrically charged, spacetimes with positive cosmological constant, called Kerr-Newmann-de-Sitter black holes, which are exact solutions of the Einstein equations. The main question we adress is the following: can we determine the metric of such black holes by observing waves at the "infinities" of the spacetime? Precisely, the considered waves will be massless Dirac fields evolving in the outer region of Kerr-Newman-de-Sitter black holes. We shall define the corresponding scattering matrix, object that encodes the far field behavior of these Dirac fields from the point of view of static observers. We finally shall show that the metric of such black holes is uniquely determined by the knowledge of this scattering matrix at a fixed energy. This result was obtained in collaboration with François Nicoleau (Université de Nantes).
*

Annalisa Panati, McGill University.

*
We consider the Nelson model with variable coeffcients, which can be seen as a model describing
a particle interacting with a scalar field on a static space time. We consider the problem of the
existence of the ground state, showing that it depends on the decay rate of the coeffcients at infinity.
We also show that it is possible to remove the ultraviolet cutoff, as it is in the flat case.
We'll explain some open conjecture.
(joint work
with C.Gérard, F.Hiroshima, A.Suzuki)
*

Scott Rodney, Cape Breton University.

* This talk will focus on functional properties of generalized Sobolev spaces and their connections to degenerate
elliptic partial differential equations. To begin, generalized Sobolev spaces, and weighted degenerate Sobolev spaces will
be defined and their connections to degenerate PDEs discussed. Following this, a general embedding result associated to
such spaces will be presented in the context of important examples. Examples to be discussed involve degeneracies encoded
by use of Muckenhoupt $A_p$ weights and also degenerate
elliptic PDEs defined with respect to collections of Lipschitz vector fields.
*

Damir Kinzebulatov, Fields Institute.

* The property of unique continuation (UC) of solutions of a PDE plays a fundamental role in
Analysis, as well as its applications to Mathematical Physics and Geometry (e.g. for proving
that Schroedinger operators do not have positive eigenvalues). The UC property is:
"if a solution vanishes on an open subset (of its connected domain) then it must be equal
to zero everywhere".
We prove UC property for solutions of the differential inequality
|\Delta u| \leq |Vu|
for V from a wide class of potentials (including "locally in L^{n/2}" class) and u in a space of
solutions Y_V containing all eigenfunctions of the corresponding self-adjoint Schroedinger
operator, extending the classical results by D. Jerison, C. Kenig [Ann. Math 1984] and
E. Stein [Ann. Math,1984].
*

Emily Dryden, Bucknell University.

* The examples of isospectral non-isometric ``drums'' constructed by Carolyn Gordon, David Webb, and Scott Wolpert show that one cannot hear the shape of a piecewise smooth planar domain $D$. They also tell us that the eigenvalues of the Dirichlet Laplace operator acting on smooth functions on $D$ form an incomplete set of geometric invariants, and it is therefore natural to look for ways to distinguish such non-isometric sound-alike drums. We will discuss what we can learn from heating these drums and studying the amount of heat in them over time. This is joint work with Michiel van den Berg and Thomas Kappeler.
*

Alexandre Girouard, Université Laval.

* In this talk, I will describe a recent joint work with L. Parnovski, I. Polterovich and D. Sher in which we have obtained precise asymptotics for the Steklov eigenvalues on a compact Riemannian surface with boundary. This has lead to a proof that the number of connected components of the boundary, as well as their lengths, are invariants of the Steklov spectrum. The proofs are based on pseudodifferential techniques for the Dirichlet-to-Neumann operator and on a number-theoretic argument.
*

© 2013 Alina Stancu